Schrodinger's Bat

Organized Common Sense

"[Economics] is a method rather than a doctrine, an apparatus of the mind, a technique of thinking which helps its possessor to draw correct conclusions."
--John Maynard Keynes, as quoted in the introduction to J.C. Bradbury's The Baseball Economist: The Real Game Exposed

Diamond Dollars deserved a review all its own, so this week we'll take a romp through The Baseball Economist to see how it stacks up.

Exposing the Game

An economist and professor at Kenesaw State University and the University of the South, Bradbury is the force behind the blog Sabernomics, where he mixes sabermetrics with economic assumptions about human behavior. Those familiar with the blog will no doubt feel right at home with the book, since many of the ideas for the chapters began as discussions and debates with his readers.

The book begins with on-field issues like hit batsmen and managers lobbying for balls and strikes, then moves on to "almost off-the-field" issues such as steroids and the evolution of baseball talent, and finally delves into front office strategies in player evaluation. He finishes by discussing issues relating to the structure of the game itself, focusing on baseball's monopoly power. The end features 75 pages of appendices that give a short overview of multiple regression, and a team-by-team accounting of the gross Marginal Revenue Product (MRP) measure that he develops earlier in the book.

The book has to walk a tightrope. On the one hand, it aims to explain economic concepts to baseball fans. At the same time, Bradbury shares the results of more hardcore sabermetric analyses related to those concepts. I found the discussions of the economic theories especially interesting as they relate to baseball. If you're just a little fuzzy on externalities, rent seeking, game theory, the Nash equilibrium, creative destruction, constrained maximization, and the difference between single and multi-price monopolies, then you're in the right place. Bradbury does a nice job of defining and explaining each idea while keeping the focus on baseball, avoiding the trap of falling into long-winded and technical explanations. Where he does diverge from baseball, he does so very briefly with the simplest of examples, in order to drive home a point. His discussion of game theory and the relationship of the Prisoner's Dilemma to performance-enhancing substances is but one of several cases that is well-handled. If only my Econ 101 professor had used baseball in his lectures, I might have actually retained something from the course.

The Studies

There are a variety of sabermetric studies in the book, and are primarily concentrated in the first two sections and include the following:

Hit Batsmen: A topic near and dear to my own heart. I referenced some of the work that Bradbury and his colleague David Drinen published in my threepartseries last summer. For those of you familiar with those articles, you won't find much really new, but the chapter is a succinct summary of the topic framed by the theories of economics.

Protection: The first section presents a study on protection (an externality) where Bradbury and Drinen find that better hitters on deck not only influence the current hitter in that they walk less frequently, but counter to the common wisdom, they also suppress batting average and power. Bradbury and Drinen explain this result with the theory that a pitcher's fear of a good on-deck hitter not only causes them to throw more pitches in the strike zone but also to expend more effort, thereby decreasing the production of the hitter being 'protected.' This study was also published in a more academic form. Although they find that the results are statistically significant, the magnitude of the effect is small, and amounts to just a few percentage points of batting average or slugging percentage for every 100-point increase in on-deck hitter OPS.

Balls and Strikes: In what is perhaps the most interesting study in the book, Bradbury looks at the effect that managers may have on umpires as they engage in rent-seeking behavior. The study analyzes QuesTecresults from 2002-2003 by comparing the strikeout-to-walk ratios of both hitters and pitchers in games inside and outside QuesTec parks, and considering away games only to remove the possible bias of home crowds. The idea is that if a manager were influencing an umpire, his pitchers would enjoy a higher ratio and his hitters a lower ratio than they would otherwise.

What he finds is that in the two sets of over 30 measures, only three results were statistically significant, indicating that the manager had some control over the calling of balls and strikes. He then ascribes the ability to influence umpires to Tony La Russa (for example), but fails to note that given his 70 observations he should expect a certain percentage of low-probability outcomes.

It turns out the number that he got (three managers) was about what one would expect with his test of significance, even if there were no actual ability by any manager to influence balls and strikes. When viewed this way, the study supports the idea that managers in fact have no influence on strikeout-to-walk ratio, and that the effort put into "managerial rent-seeking" might be better used in other pursuits. An alternative hypothesis--difficult to test, of course--is that rent-seeking behavior by managers is a game of push and pull, one that keeps umpires on the level, resulting in no evidence of an effect either way. Although it's unlikely that it would matter, it would be interesting to see this study taken to another level by drilling down to the pitch level and looking at the ratio of called strikes to balls. The enhanced GameDay data provided by MLBAM would be a nice place to start.

The Mazzone Effect: One of the studies that Bradbury is surely most well-known for is his look at the effect erstwhile Braves pitching coach Leo Mazzone had on pitchers who came to the Braves. Since the study is also summarized elsewhere, I won't describe it in detail. Suffice it to say that he found a large impact, or -0.640 earned runs per nine innings. In other words, Mazzone's influence was good for more than a half-run per game when averaged between starters and relievers, with relievers appearing to benefit more than starters. He also breaks it down by strikeout, walk, and home run rates, and shows that while starters saw their strikeout rates increase and walk rates decrease more than relievers, relievers improved their home run rate more than starters.

While I don't doubt that Mazzone had some effect because of the pitching philosophy he learned from Johnny Sain and passes along today, I find it difficult to believe that we can ascribe the majority of an improvement of that magnitude to one person. Bradbury admits to that possibility, noting that the Braves management was also extraordinarily stable during Leo's tenure. That organizational effect is perhaps hinted at because, as Bradbury also documents, Braves pitchers did not take their improvement with them once they left Leo's tutelage. In either case, a couple more years of data in Baltimore should help make the point--or not.

Big and Small: In this study, Bradbury goes to some length to argue that market size is not destiny. He suggests that it isn't the case that "small markets are doomed to perpetual failure." To demonstrate this he correlates market size (using the 2000 census data) with average wins from 1995 through 2004, and then concludes that 40 percent of the difference in wins from top to bottom can be credited to market size, leaving 60 percent to management. After adjusting wins for population and attempting to calculate how many playoff appearances teams lost by virtue of their market (where every 1.58 million people generate an additional win), he concludes:

While big-market teams may have an advantage over small-market teams, the advantage appears to be slight and virtually meaningless. The bigger problem appears to be inept management of a few clubs that happen to be small-market teams.

There is no doubt that several small market teams have shot themselves in the proverbial foot in recent years, but still I can't help but think that Bradbury is attempting to minimize the link between market size and success on the field. Using Nate Silver's recently published rankings of attendance sphere and TV sphere, along with average payroll, metropolitan statistical area (MSA) and the Mike Jones ratings, I took a crack at the correlations of each with average wins, average division ranking, and number of postseason appearances from 2002 through 2006. The coefficient of determination (which can be interpreted as the percentage of the difference between teams that can be explained by the metric) for each is shown in the table belowL

Using only Silver's Attendance sphere, the correlation appears to be somewhat stronger than what Bradbury found. It accounts for over 60 percent of the difference in wins between the top (Yankees) and bottom team (Royals). It would seem that a 20-win difference is in fact not a "meaningless" difference, but one that yields a substantial alteration in the probability of a team reaching the postseason. This fact is illustrated by looking at the average Attendance Sphere of division finishers since 2002.

As discussed by Vince Gennaro in Diamond Dollars, the advantage of large markets is compounded by the fact that once a team reaches the postseason they can expect an additional 8.5% of baseline revenues in the first year following their postseason appearance, thanks to an increased level of fan excitement (often accompanied by rising ticket prices). For example, the 2005 White Sox were estimated to receive a bonus of $17 million in 2006 based on their postseason run. That benefit continues for the next several years, reaching approximately $28 million according to Gennaro's model. Although the effect dissipates after multiple consecutive trips to the postseason, the advantage of market size is compounded in a kind of a positive feedback loop that keeps the benefits accruing.

What's more interesting from the preceding table, though, is that both TV sphere and payroll are more strongly correlated with winning than market size is (either the attendance sphere or MSA). The graphical representation of the correlation of payroll and wins is shown below.

So while market size is clearly an important factor, commanding a larger TV audience and its associated revenue before shelling out that revenue for player salaries is even more significant in terms of on-field success.

Judging Hitters and Pitchers: Again, it's difficult to write a book for a general audience and readers with a background in sabermetric analysis. The latter can probably skip the chapter in section three where Bradbury walks through both traditional and non-traditional ways of evaluating performance. That said, if you want to give a short primer to someone on the reasons batting average and ERA are not the best of measures of individual performance, or if you want to introduce the usefulness of DIPs theory, this chapter is perfect.

Evolving Talent: In this short and thought-provoking chapter, Bradbury argues that the increasing offensive levels of recent years are not necessarily only (or even primarily) due to the influence of performance-enhancing drugs, but instead the dispersion of talent at the major league level. Using Stephen J. Gould's theory that decreasing variation means increasing ability, he measures the dispersion of hitting and pitching using OPS and ERA by decade, and shows that pitching variability is at an all-time high driven (in Bradbury's estimation) by the latest two rounds of expansion in 1993 and 1998. Hitting variability has also increased, but to a lesser extent. More inferior pitchers in the game are, at least in part, responsible for the higher offensive levels and the feats of sluggers like Mark McGwire, Sammy Sosa, and Barry Bonds. He reiterates this argument in a recent New York Times piece, and it's an idea that I've certainly been chewing on.

Worthy Ballplayers

With the more traditional sabermetric studies out of the way, Bradbury moves on to measuring the value of players using his Marginal Revenue Product (MRP) metric. He calculates this by first estimating the impact of runs on revenue using run differentials. This allows him to draw a non-linear curve not entirely dissimilar from the win-revenue curve that Nate Silver discusses in "Is Alex Rodriguez Overpaid?" from Baseball Between the Numbers.

Bradbury finds that the first run scored or prevented that puts a team above .500 is worth $127,000. It turns out that at a .500 win percentage Bradbury's figures are pretty close to what Silver found, where the 82nd win was worth about $1.2 million. Roughly ten runs equate to one win, so the first win above .500 would be worth around $1.3 million. However, in Silver's more sophisticated model, the wins from 83 to 90 bring in increasingly more revenue, with the 90th win worth $3.5 million. After 90, each additional win brings in less revenue because of its decreasing relevance in making the playoffs. In Bradbury's model, the curve is less steep, though additional revenue continues to accrue to around $175,000 per run as the run differential nears 200, equating to just over 100 wins.

Silver's general model is supported by Gennaro's analysis in Diamond Dollars, where his win-revenue curves look very similar. Gennaro takes the additional steps of customizing them per team, and augmenting the curves for additional postseason and World Championship revenue. But since Bradbury is using this analysis to get at the dollar value of players and not trying to predict the impact on revenues if a particular team adds a particular player, his more linear approach is better suited for his next steps.

Bradbury next calculates the number of runs a hitter would produce, and the number of runs a pitcher saves, if they were a team unto themselves. He uses formulas derived from regression equations, considering OBP and SLG for hitters, and strikeout, walk, and home run rates for pitchers (taking DIPs theory into account). These values are then park-adjusted and prorated by playing time to produce a Runs Scored Above Average for hitters (RSAA) and runs allowed below average (RABA) for pitchers.

Finally, these values are translated into revenue figures using the curve he created in step one, added to a baseline amount of revenue the average player would contribute given the same percentage of playing time (with "average player" defined as one who would produce an 81-81 record and thereby generate $54.5 million in revenue). The result is an MRP value for each player in 2005 and 2006: Derrek Lee took the top spot in 2005 at $19.18 million, and Albert Pujols lead with $20.58 million in 2006.

While this technique allows Bradbury to create a single list of players with a value attached to each, it overlooks the more complex reality that a Derrek Lee who produced 74.38 RSAA in 2005 is actually more valuable to a team who capitalizes on his additional run differential to move into the 90-win territory, thereby increasing revenue in non-linear way, as per the Silver and Gennaro win-revenue curves. In the market for free agents, players are often viewed in terms of the "last piece of the puzzle" in Gennaro's terms; teams are, or should be, willing to ante up commensurate with their predicted place on the win-revenue curve given that player's contribution(s). As a result, what a player is really worth depends in great deal on the teams that are interested in signing him.

Secondly, Bradbury's technique doesn't take into consideration the defensive value of a player, nor is it based on replacement value like Gennaro's system, which uses WARP.

These two problems compound one another, skewing the one-size-fits-all valuations of some players. Take Bradbury's example of Neifi Perez. In 2005 Perez consumed just under 10 percent of the Cubs' plate appearances; in this model, if he were an average hitter, he would have been valued at $5.45 million. Since he was in fact below average (-16 RSAA), he is valued at $3.54 million. Remember that this method is based only on offensive performance, and in defending it, Bradbury argues that the valuation is fair since Dusty Baker played Perez because "he brings something to the team other than his offense." In other words, Neifi's other contributions are rolled into the value as evidenced by his playing time. Perez's defense was exemplary in 2005, and rated a +20 in The Fielding Bible .

But Bradbury's system doesn't account for defense. The fact that Perez's overall performance (a 4.3 WARP) may be in the ballpark with his calculated MRP is really more of a coincidence. What if Perez had been an average defender along the lines of Felipe Lopez or David Eckstein, but Baker simply had no other choice but to play him? In that scenario, it would be clear that his value was nowhere near three and half million dollars, and was instead closer to the replacement level salary of $380,000. In fact this was precisely the case with Angel Berroa, who in 2005 had a -17.7 RSAA and registered a -26 in the Plus/Minus system, and yet is valued by Bradbury at $3.82 million. If Bradbury had used a replacement level baseline (and included defense) he could have still used his curve to assign player valuations (starting at a much lower baseline in terms of revenue for each additional run). The result would be much more realistic results for players like these.

The Big Picture

Despite the difficulties with MRP, Bradbury then explains how the valuations are related to actual salaries. Here he does a nice job of explicating the differences in player classes, and how those differences manifest themselves in value versus salary. He also discusses where the additional revenue not paid out in salaries is spent.

In the final section, Bradbury is clearly in his element, discussing major league baseball's monopoly power (and how it relates to other sports leagues), the shape that monopoly takes, and how market forces have acted on the league to bring "fans the baseball we deserve." His lucid descriptions of single- and multi-price monopolies--and his argument that baseball acts as the latter, thereby leading to more games at acceptable prices for fans--seem well-reasoned. He then couples this with an argument that baseball is a contestable market, as evidenced by the Federal League and the threat of the Central League that forced the industry through an "invisible hand" to keep expanding as the country grew. This section alone makes the book worth reading.

The wide-ranging subject matter and the connection of baseball to economic concepts far overshadow the minor quibbles I have with it. And since that was one of the primary goals Bradbury had in mind, I'd call the effort a success and recommend the book for those seeking a challenge.